This is another update of my answer, in which I also give a quick summary about solvability obstructions as suggested in Will Sawin's answer.

The smallest equation with easy-to-find solution is $0=0$ with $H=0$. The smallest equations with no solutions are $1=0$ and $-1=0$ with $H=1$. The smallest equations with at least one variable and no solutions are $x^2+1=0$ and $2x+1=0$ (and their variants) with $H=5$.

All equations with $H\leq 14$ either have small solutions or have trivial obstructions of at least one of the following types: 
a) no real solutions (like $x^2+1=0$) or no real solution outside a region with finite number of integer points (like $x^2-2=0$), 
b) no solutions modulo some integer (like $2x+1=0$), or
c) divisibility conditions imply at most finitely many possible solutions, and none of them works. An example is $(x^2+2)y=1$, where $y$ must be a divisor of $1$, but both divisors $y=1$ and $y=-1$ do not lead to a solution.

Equation $y^2=x^3-3$ with $H=15$ is the smallest equation with no solutions for which the trivial reasons above do not suffice and obstructions arising from divisibility properties of values of quadratic forms are needed. This equation belongs to the family of Mordell's equations and is well studied. The smallest equation with this type of obstruction which does not seem to be well-studied is $y(x^2-1)=z^2+1$ with $H=17$. This analysis covers all the equations up to $H\leq 21$.

Equation $xyz=x^2+y^2-z^2+2$ with $H=22$ is the smallest one with obstructions from Vieta jumping. This equation has been recently solved by Will Sawin and Fedor Petrov. 

The listed obstructions suffice to solve all equations up to $H\leq 25$. There are several equations with $H=26$ which I currently do not see how to solve. An example is
$$
y(x^3-y)=z^2+2.
$$ 

**Update**: This equation is now solved. For the solution and the next smallest open equations see https://math.stackexchange.com/questions/4159235/is-this-small-diophantine-equation-solvable?noredirect=1#comment8722722_4159235